Banco de México

Working Papers

N° 2021-16

Forward Guidance in an Advanced Small Open

Economy in the Effective Lower Bound

October 2021

La serie de Documentos de Investigación del Banco de México divulga resultados preliminares de

trabajos de investigación económica realizados en el Banco de México con la finalidad de propiciar elintercambio y debate de ideas. El contenido de los Documentos de Investigación, así como lasconclusiones que de ellos se derivan, son responsabilidad exclusiva de los autores y no reflejannecesariamente las del Banco de México.

The Working Papers series of Banco de México disseminates preliminary results of economicresearch conducted at Banco de México in order to promote the exchange and debate of ideas. Theviews and conclusions presented in the Working Papers are exclusively the responsibility of the authorsand do not necessarily reflect those of Banco de México.

Marine Char lo t te AndréBanco de México

Guido Traf icanteUniversità Europea di Roma

Forward Guidance in an Advanced Small Open Economyin the Effect ive Lower Bound*

Abstract: We examine forward guidance (with known and uncertain duration) in a New Keynesianmodel for an advanced small open economy, showing that the response of the economy to this policydepends, both quantitatively and qualitatively, on some structural features through calibrations forSweden and Spain. In particular, an announcement of future expansionary policy is positively related tothe exchange rate pass-through and is larger than in the closed economy counterpart because of a betterinflation-output trade-off and the exchange rate channel. We also show that multiple equilibria couldarise and that the real exchange rate is a key variable driving this result. In particular, the response ofoutput and inflation is amplified when aggregate supply is negatively related to the real exchange rate.These results could not necessarily be extended to emerging market economies.Keywords: Monetary policy, advanced small open economy, forward guidance.JEL Classification: E31, E52

Resumen: Estudiamos la guía futura (con duración conocida e incierta) en un modelo Nuevo-Keynesiano para una economía avanzada pequeña y abierta, mostrando que la respuesta de la economíaa esta política depende, de manera cuantitativa y cualitativa, de algunas características estructuralesmediante calibraciones para Suecia y España. En particular, anunciar una política expansiva futura estárelacionado positivamente con el traspaso del tipo de cambio a la inflación y es mayor que en unaeconomía cerrada, debido a una mejor disyuntiva entre producto e inflación y al canal del tipo decambio. Además, se demuestra que múltiples equilibrios pueden ocurrir y que el tipo de cambio real esuna variable crucial para obtener tal resultado. En particular, la respuesta del producto y la inflación seamplifica cuando la oferta agregada está relacionada negativamente al tipo de cambio real. Estosresultados no necesariamente se podrían extrapolar a economías emergentes.Palabras Clave: Política monetaria, economía avanzada pequeña y abierta, guía futura.

Documento de Investigación2021-16

Working Paper2021-16

Marine Char lo t t e André y

Banco de MéxicoGuido Tra f ican te z

Università Europea di Roma

*We thank Jacopo Bonchi, Giorgio Di Giorgio, Fausto Gozzi, Serdar Kabaca, Nigel McClung, SalvatoreNisticò, Alexander Scheer, all the participants at CASMEF seminar and at Western Economic Association VirtualInternational Conference for their comments.Agradecemos a Jacopo Bonchi, Giorgio Di Giorgio, Fausto Gozzi, Serdar Kabaca, Nigel McClung, SalvatoreNisticò, Alexander Scheer, los asistentes al seminario CASMEF y a la conferencia Western EconomicAssociation Virtual International por sus comentarios. y Dirección General de Investigación Económica. Email: mandree@banxico.org.mx. z Università Europea di Roma, Rome, Italy. Email: guido.traficante@unier.it.

1 Introduction

The global financial crisis has triggered a vivid interest in theoretical and empirical re-

search on unconventional monetary policies that look for a substitute of the short-term

nominal rate when the latter reaches the zero lower bound. A key example of such uncon-

ventional policies is given by forward guidance, through which policymakers announce

a path of the nominal interest rate starting immediately or in the future for a particular

duration. Through this policy, the central bank tries to manage expectations of the future

policy rates once the zero lower bound is no longer binding in order to influence macro-

economic dynamics. In the basic New Keynesian DSGE model, an anticipated change in

the real rate produces an effect on output which is independent of the duration and timing:

this is the forward guidance puzzle. As a consequence, the effects of a temporary vari-

ation in the policy rate that takes place very far in the future is the same if the variation

were to take place immediately or in the near future. This puzzle, discussed by Del Negro

et al. (2012), Carlstrom et al. (2015) and McKay et al. (2016), derives from the fact that,

in a baseline New Keynesian DSGE model, the dynamic IS relationship has no discount-

ing of the expected output gap and, in turn, of future real interest rates. Consequently,

the literature introduced some discounting mechanism in the Euler equation so that ag-

gregate demand responds less than one-to-one to its future expected changes. Examples

include an overlapping-generations structure a la Blanchard and Yaari in the demand side

(Del Negro et al. (2012)), heterogeneous agents and incomplete markets (McKay et al.

(2016)), sticky information (Carlstrom et al. (2015)), bounded rationality (Beqiraj et al.

(2019)). McClung (2020) shows that a regime characterized by passive monetary policy

and active fiscal policy does not imply forward guidance puzzle. With active fiscal policy,

Ricardian equivalence does not hold and agents perceive government debt as net wealth.

As a consequence, forward guidance announcements that lower the expectations of fu-

ture interest rates produce negative wealth effects that counteract the monetary stimulus.

Furthermore, Hagedorn et al. (2019) show that under commitment, forward guidance has

only transitory effects on the economy and Nakata et al. (2019) study optimal forward

guidance in a setup where forward guidance puzzle is attenuated.

In this paper we analyze the theoretical implications of forward guidance in an ad-

vanced small open economy, focusing on the international transmission of such a policy.

To the best of our knowledge, Galı (2020) is the only theoretical contribution about for-

ward gui-dance in open economy. He shows that if the home central bank announces

an increase (decrease) of the nominal interest rate of T periods, with no reaction from

the foreign central bank, the exchange rate appreciation (depreciation) at the time of an-

1

nouncement is proportional to the duration and the size of the interest rate change, but

it is independent on the duration of the forward guidance. Therefore forward guidance

puzzle arises also in a small-open economy model. We follow Galı and Monacelli (2005)

and Leitemo and Soderstrom (2008) in modeling a small country that freely trades with

the rest of the world, constituted of a continuum of foreign economies. Forward guidance

will be analyzed in a deterministic scenario, where its duration is known with certainty,

and in a stochastic setting, modeled along the lines of Bilbiie (2019). We evaluate the

forward guidance policy in normal times and under a liquidity trap with the central bank

committing to keep the interest rate fixed also when the economy is out of the liquidity

trap.

Our main results are the following ones. First, we show that forward guidance may

have distinct effects on inflation and output gap, both in terms of magnitude and direc-

tions, compared to the closed economy. A key determinant is the elasticity of inflation

with respect to the real exchange rate which could be either positive or negative. In the lat-

ter case, in a liquidity trap forward guidance may lower the output gap and inflation, thus

offsetting the stimulating intertemporal effects of expansionary forward guidance when

the forward guidance policy is followed for a short-period of time1. Second, the real

depreciation that occurs in presence of negative elasticity of inflation with respect to the

real exchange rate drives the expansionary effects of forward guidance also in a stochas-

tic setup, where the duration of the policy and the state of the economy (“normal times”

versus liquidity trap) follow a Markov chain. We also show that keeping the interest rate

fixed after the liquidity trap is over can generate an increase in output gap and inflation

instead in an advanced small open economy. Finally, compared to the closed-economy

counterpart, forward guidance tends to be more expansionary in open economy: this is

due to the combination the role played by the real exchange rate and to the better trade

off between output and inflation (because of a larger Phillips’ curve slope). Such results

might not necessarily be applied to emerging market economies.

The paper is organized as follows: Section 2 describes the reference model, while

Section 3 studies forward guidance when the duration of the policy is known with cer-

tainty. In Section 4 we consider the same experiment with uncertainty about the duration

of liquidity trap and forward guidance before concluding in Section 5.

1The transmission under positive pass-through proceeds inversely in the liquidity trap.

2

2 The Model

We shortly summarize, with some slight changes in notation, the small open economy

model of Galı and Monacelli (2005) and Leitemo and Soderstrom (2008). With the ob-

jective of deriving analytical solutions, the only shock (defined later) is a preference shock

that drives the economy into a liquidity trap.

The small domestic country freely trades with the rest of the world (foreign country),

constituted of a continuum of foreign economies. We assume that foreign and domestic

countries share preferences and technology. Domestic and foreign firms produce traded

consumption goods, using labor as the sole input. Households derive their utility from

consuming both domestic and foreign goods, and have a marginal decreasing disutility in

labor supply to firms.

Denoting by et the log-linearized real exchange rate, we have by definition

et = st + p ft − pt , (1)

with st being the nominal exchange rate (units of domestic currency against one unit of

foreign currency), p ft the price level of the goods produced in the foreign country and pt

the price level of domestically produced goods.

The real exchange rate is directly related to the inflation rate in the domestic goods

sector, πt , via the New Keynesian Phillips curve:2

πt = βEtπt+1 +κxt−φet , (2)

where xt denotes the output gap, 0 < β < 1 the discount factor, and Et the rational expec-

tations formed by private agents (conditional on information set available at time t). The

composite parameter κ = κ(η +σ), with κ ≡ (1−ϑ)(1−ϑβ )ϑ

, is the output-gap elasticity of

inflation and encompasses the effect of the output gap on inflation via real marginal costs.

Phillips’ curve slope depends on ϑ , the share of firms that do not optimally adjust but sim-

ply update in period t their previous price by the steady-state inflation rate, on η , which

represents the steady-state Frisch elasticity of labor supply, and on σ ≡ σ

1−ωwith σ denot-

ing the inverse of the elasticity of intertemporal substitution, and 0 ≤ ω ≤ 1 the share of

foreign goods in domestic consumption. The real exchange rate enters the Phillips curve

2Differently from Galı and Monacelli (2005), Leitemo and Soderstrom (2008) derive a Phillips curveincluding the real exchange rate. For the microfoundations of the model, see Leitemo and Soderstrom(2008). Notice that πt is different from the inflation rate of the consumer price index that also takes intoaccount the inflation of foreign goods consumed by residents. In the closed economy, πt represents bothproducer and consumer price inflation rates.

3

through the coefficient φ = ωκ [(2−ω)ζ σ −1], where ζ stands for the elasticity of sub-

stitution across domestic and foreign goods. In general, there is not unanimous consensus

about the sign of the relationship: differently from Leitemo and Soderstrom (2008), Walsh

(1999) and Razin and Yuen (2002) obtain a positive relationship between these two vari-

ables in theoretical models. Therefore, in our analysis we will consider both signs in the

relationship. The economic intuition behind the relationship between inflation and real

exchange rate is the following: when households choose labor supply, they care about the

purchasing power of their wage deflated by the consumer price index that also includes

prices of imported goods, implying that the equilibrium wage and hence the real marginal

costs depend on the real exchange rate. As highlighted in Leitemo and Soderstrom (2008),

there are two competing effects shaping the relationship between exchange rate and in-

flation. On the one hand, an exchange rate depreciation (i.e. an increase in the exchange

rate) increases consumer prices and therefore reduces households’ purchasing power. The

optimal labor supply choice will imply higher wages and, in turn, higher marginal costs

and inflation. On the other hand, an exchange rate depreciation leads to a decrease in the

demand for imports and therefore a reduction in aggregate consumption. The marginal

rate of substitution then falls, leading to lower real wages and marginal cost. The compos-

ite parameter φ is positive as long as (2−ω)ζ σ > 1: this condition holds in Leitemo and

Soderstrom (2008), determining a negative relationship between inflation and exchange

rate for their model calibrated to Sweden. However, for economies whose main exports

are based on price competitiveness, generally the first effect dominates and an exchange

rate depreciation induces higher inflation, which reduces domestic consumption. For in-

stance, Mihailov et al. (2011) show that for Spain a currency depreciation increases the

possibility to export at the cost of a lower purchasing power for consumers.

The New Keynesian IS equation is given by

xt = Etxt+1−σ−1(rt−Etπt+1)+σ

−1ρt−δ (Etet+1− et) , (3)

where rt is the nominal short-term interest rate, ρt represents an exogenous disturbance

that moves the natural interest rate, and δ a composite parameter defined by δ ≡ 1σ

[Ω

(1−ω) −1]

with Ω ≡ (1−ω) [(1−ω)+(2−ω)ωζ σ ]. The composite parameter δ is the elasticity

of the output gap with respect to the expected change in the real exchange rate, reflecting

the substitution effect induced by such a change on the demand of domestically produced

goods.3 Also with respect to the output gap, there are two competing effects of exchange

rate movement. On the one hand, an exchange rate depreciation increases consumer prices

3Note that Ω and δ are positive for (2−ω)ζ σ > 1.

4

and reduces expected inflation; the resulting increase in the real interest rate reduces con-

sumption and the output gap, given the expected future exchange rate. On the other hand,

the exchange rate depreciation increases export demand, and therefore output. As shown

in Leitemo and Soderstrom (2008), the same condition shown above for φ determines the

type of relationship between exchange rate and output gap indirectly through the Phillips

curve; φ determines whether a country would export more following a depreciation of its

national currency.

Finally, the real UIP condition relates the real interest rate differential with the ex-

pected rate of real depreciation:

rt−Etπt+1 = Etet+1− et , (4)

where foreign variables are set to zero for simplicity.

3 Forward Guidance with Known Duration

In this section, we consider the case in which the central bank announces that it will keep

the nominal interest rate fixed for T periods. The central bank is assumed to conduct

policy through a Taylor rule like the following one4

rt = ρt +φππt φπ > 1 (5)

Such a rule guarantees that ignoring the zero lower bound there is a unique equilibrium

characterized by πt = xt = et = 0.

The experiment that we consider is the following. After a shock to the natural rate the

economy enters in the zero lower bound for a period of time. The central bank responds by

adopting forward guidance, ie announcing that the nominal interest rate will be constant

to r for a number of quarters T that does not necessarily coincide with that of the liquidity

trap. We characterize the dynamics of the economy under the forward guidance period

knowing that, after the forward guidance regime, the Taylor rule will provide a unique

equilibrium. In particular, at time T , inflation, output gap and the real exchange rate will

4The specification of the Taylor rule is such that in normal times equilibrium is unique. Alternativespecifications, including a response to output gap and/or real exchange rate could be considered.

5

be equal to

πT =(

φ −κδ − κ

σ

)r

xT =−(

δ +1σ

)r

eT =−r

To solve the model during forward guidance, we write the system (2) – (4) in matrix

notation with Z ≡(

π x e)′

. It is convenient to invert this system, running time

backwards from the end of the period of fixed interest rates. That is, let Ys denote the

value of Z s periods before time T : Ys ≡ ZT+1−s. Changing the system in this way will

allow us to reinterpret the final conditions on π , x and e as initial conditions in a system

of difference equations. Therefore the system (2) – (4) is written as

Ys = AYs−1 +Br (6)

where A≡

β + κ

σ+ κ

δ−φ κ −φ

1σ+δ 1 0

1 0 1

, B≡

−κ( 1

σ+δ)

− 1σ−δ

−1

The system (6) has a solution composed by a non-homogenous part and a homoge-

neous part, while the initial values will be given by the vectorφ − κ

σ− κ

δ

− 1σ−δ

−1

r

In this model, it turns out that the solution is not unique, as the matrix (I−A) is not

invertible and the system admits ∞1 solutions. This result confirms the classical result

of exchange-rate indeterminacy following exogenous interest-rate rules, even under the

assumption that this policy has a finite duration, which has been extensively discussed in

the literature, as in Kareken and Wallace (1981), Obstfeld et al. (1996) and Benigno et al.

(2007). However, the system admits solutions with the following structure

YT = Z1 = ATY0 +T−1

∑s=0

AT−s−1Br (7)

The previous model can be solved analytically but the results are hard to interpret.

Therefore we calibrate it and solve it numerically. As a baseline calibration, we follow

the values that Corbo et al. (2020) use for Sweden. In particular, they assume that σ = 1,

6

as well as the elasticity of substitution between domestic and foreign goods ζ . The Frisch

elasticity of labor supply η = 3.65, while the share of firms that do not optimally set

their prices ϑ is 0.93 and the degree of openness ω = 0.19. These values imply that

the composite parameters are given by κ = 0.0292, σ = 1.235, φ = 0.0011 and δ = 0.15.

Therefore, this calibration is related to a country where the real exchange rate is negatively

related with inflation and output.

0 2 4 6 8 10 120

0.5

1

1.5Initial response as a function of T - open economy

0 2 4 6 8 10 120

0.2

0.4Initial response as a function of T - closed economy

inflationoutput

0 2 4 6 8 10 12T

0

0.5

1

1.5

real

exc

hang

e ra

te

Figure 1: Initial response of the economy: closed versus open economy (authors’ Matlabsimulation)

Figure 1 compares the initial response of inflation and output gap during the forward

guidance regime in open economy (top panel) and in closed economy (mid panel) as

a function of forward guidance duration. The graph shows that the forward guidance

puzzle arises both in closed and open economy and that the response of inflation and

output gap is magnified in open economy5 especially if we increase the time in which

interest rate is fixed. We believe that two factors determine this higher response in open

economy. The first one is the larger interest rate elasticity of the output gap which, in

turn, improves the inflation-output trade-off in the Phillips’ curve. The second factor is

the exchange rate depreciation which boosts aggregate demand; in particular the exchange

rate (bottom panel) moves exactly in the same way as the output gap6 and this reinforces5This is the reason why we put in two separate panels the variables in closed versus open economy.6This is due to the fact that, as we showed before, in open economy, the system (2) – (4) admits infinite

solutions so that output gap and exchange rate cannot be decoupled.

7

the expansionary effect of forward guidance.

Figure 2: Initial response of the economy with negative φ and δ (authors’ Matlab simu-lation)

As highlighted in section 2, the elasticity of inflation and output with respect to the

real exchange rate can be either positive or negative. Therefore, we repeat the same

analysis as before calibrating the model following the evidence Mihailov et al. (2011) for

Spain. More in detail, the different values for such an economy are ζ = 0.25, σ = 0.78,

ϑ = 0.85 and ω = 0.25, which, in turn, imply σ = 1.045, κ = 0.1131, φ = −0.0038

and δ = −0.1299. Figure 2 shows that, with negative passthrough, there are two stark

differences compared to the previous analysis. First of all, it shows that the response of

the economy is lower to both the cases described in Figure 1: in this case, due to the lower

reactiveness of inflation and output gap to the exchange rate, inflation, output gap and the

exchange rate react much less. Moreover, the initial level of inflation is negative and it is

necessary to extend the duration of forward guidance (not shown here) to have positive

inflation. This suggests that if a central bank wants to stimulate the economy through a

forward guidance policy, it is necessary a commitment to maintain this regime for a long

period when there is a low reactiveness to the exchange rate. Also in this case, under a

theoretical viewpoint, there is forward guidance puzzle, albeit on a lower scale.

8

Figure 3: Response of the economy varying the elasticity of substitution between do-mestic and foreign goods. Inflation and output gap are expressed as a ratio over theirclosed-economy counterparts (authors’ Matlab simulation).

The previous analysis has been conducted under the assumption of a unitary elasticity

of substitution between domestic and foreign goods. However, over this parameter, there

is not unanimous consensus in the literature. As emphasized by Di Giorgio and Nistico

(2013), when using aggregate data, the literature provides evidence of a value for the

elasticity in the range of unity, as in Hooper et al. (2000), while this value is much above

one when disaggregated data are used (see Obstfeld and Rogoff (2000) and McDaniel

and Balistreri (2003)). More recently, Bajzik et al. (2020) collect 3524 reported estimates

of the elasticity and investigate what drives the heterogeneity in the results. Taking into

account also publication bias, the elasticity lies in the range 2.5-5.1 with a median of

3.8. Based on this empirical evidence, we repeat the analysis performed above varying ζ

(and consequently φ and δ ) in figure 3 for a forward guidance duration of T = 12. The

figure shows that the initial response of the three endogenous variables are increasing in

the elasticity of substitution: this occurs because the larger the elasticity of substitution,

the larger the exchange rate pass-through to inflation and the output gap. As a result, the

effects of a policy that keeps the policy rate fixed are amplified.

9

4 Forward Guidance with Stochastic Duration

Now we consider a version of the model where the duration of forward guidance is

stochastic, focusing, as before, on the difference between closed and open economy, and

also on the role played by exchange rate passthrough.

We assume that the central bank performs a fully credible forward guidance policy.

As in Bilbiie (2019), we model forward guidance stochastically through a Markov chain

as a state of the world with a probability distribution of p for the liquidity trap to hap-

pen. Consequently the expected stochastic duration of the liquidity trap is TL = (1− p)−1

which is the stopping time of the Markov chain. ρt follows a Markov chain of 3 states,

one first state is the steady state S where ρt = ρ and once reached, there is a probability

1 of staying there. The second state is the liquidity trap, being transitory, denoted by L

where rt = 0 and ρt = ρL < 0 with persistence probability p. After this time TL, the CB

sets rt = 0 while ρt = ρ > 0, with probability q. The probability to move back to steady

state from F is 1−q. We denote this state F , with expected duration TF = (1−q)−1. The

state F can be interpreted as CB’s commitment to maintain a low policy rate low despite

the fact the economy is out of the liquidity trap. Therefore, we have the following three

states of the world:

1. Liquidity trap L, with rt = 0 and ρt = ρL. The economy remains in this state with

probability p and arrives to the state of forward guidance with probability (1− p)q.

2. Forward guidance F , with rt = 0 and ρt = ρ . The economy is in this state with

probability q and goes back to the steady state with probability 1−q.

3. Steady state S with rt = ρt = ρ (absorbing state).

Given these assumptions, we can write the following expectations for our endogenous

variables:

Etxt+1 = pxL +(1− p)qxF ,

Etπt+1 = pπL +(1− p)qπF

Etet+1 = peL +(1− p)qeF

4.1 Closed versus Open Economy

A natural question that arises is if forward guidance is more or less effective in open

economy. Figure 4 shows the value of inflation and output gap, both in state F and in

10

state L, for the open-economy case (with φ = 0.0011, as for Sweden, solid lines) and for

the closed-economy case (circled line), varying the probability of forward guidance q.

In state F , we can observe that the variables barely move when q is lower than 0.2

while they respond much more for large value of q (specifically, for q > 0.8). In terms

of output, we observe the same peak in closed and in open economy, even if in open

economy it is necessary to follow a fixed-interest rate policy for a longer period to arrive

at this point. Interestingly, in state F forward guidance is not monotonically expansionary,

in fact for q > 0.8 in open economy and for q > 0.6, we observe a deflation associated

with an appreciation and a recession.

In state L, forward guidance is more expansionary in open economy: this is due to

larger Phillips’ curve slope κ and interest rate elasticity of the output gap σ , compared

to closed economy. The other factor explaining the more expansionary effect in open

economy is the exchange rate depreciation which boosts aggregate demand without the

same increase in inflation observed in closed economy, due to the possibility of substi-

tuting domestic goods with foreign ones. Moreover, in closed economy a shorter period

of forward guidance (0 < q < 0.5) is sufficient for the economy to reach the largest ex-

pansion for output in normal times due to direct transmission channels between inflation

and output gap. Being in a small-open economy, the dynamics of the output gap follow

closely those of the exchange rate, as highlighted by the literature, eg Galı and Monacelli

(2005).

In state L, the path followed by output gap and inflation presents more differences

across open and closed economy. While in closed economy the effect is almost muted up

to approximately before q = 0.5 and then we observe a through followed by a peak, in

open economy the effect is globally more expansionary (as already discussed for state F)

and the troughs are sensibly lower. More in detail, the economy experiences a peak for

q = 0.65, then we observe a decrease with inflation and output gap going into negative

territory. Again, there is a key contribution of the real exchange rate: when it appreciates

the economy enters in a deflation and a recession.

4.2 Comparison of Different Open Economies

We now compare the effects of forward guidance duration for two different open economies

using different calibrations for Sweden and Spain respectively (see Table 1).

In the forward guidance experiment shown in Figure 5, the variables evolve in the

same direction whether the economy has a positive or a negative exchange rate pass-

through. In state F , the largest exchange rate depreciation, obtained between 0.6 ≤ q <

11

0 0.2 0.4 0.6 0.8 1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Output gap - State F

0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Inflation - State F

0 0.2 0.4 0.6 0.8 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3Exchange Rate - State F

0 0.2 0.4 0.6 0.8 1

q

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Output gap - State L

0 0.2 0.4 0.6 0.8 1

q

-1

-0.5

0

0.5Inflation - State L

Open Economy

Closed Economy

0 0.2 0.4 0.6 0.8 1

q

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5Exchange Rate - State L

Figure 4: Comparison of forward guidance between a closed and open economy (authors’Matlab simulation)

ParametersSweden

Source ParametersSpain

Source

β 0.99 Corbo et al. (2020) 0.99 Mihailov et al. (2011)σ 1 Corbo et al. (2020) 0.78 Mihailov et al. (2011)ζ 0.87 Corbo et al. (2020) 0.25 Mihailov et al. (2011)ω 0.19 Corbo et al. (2020) 0.25 Mihailov et al. (2011)ϑ 0.93 Corbo et al. (2020) 0.85 Mihailov et al. (2011)η 3.65 Corbo et al. (2020) 3 Mihailov et al. (2011)σ 1.2346 Author’s calculation 1.04 Author’s calculationφ 0.011 Author’s calculation -0.0038 Author’s calculationκ 0.029 Author’s calculation 0.1130 Author’s calculation

Table 1: Calibrations for Sweden and Spain

0.7 goes hand-in-hand with a peak in inflation and output gap for the case of a positive

exchange rate pass-through (dashed line). For the case of Spain, i.e., negative exchange

rate pass-through, we observe that we should at least engineer a forward guidance duration

of 2.5 quarters to obtain a response of the exchange rate that depreciates in a small interval

between 0.6 < q < 0.7. Also in this case, the depreciation is associated with an expansion

of output gap and inflation. Overall, the effect of forward guidance policy for Spain are

12

lower compared to Sweden because of the much lower exchange rate pass-through7.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

F

= -0.0038 = 0.0011

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

L

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.1

0

0.1

0.2

0.3

x F

= -0.0038 = 0.0011

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

x L

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

q-0.3

-0.2

-0.1

0

0.1

0.2

0.3

e F

= -0.0038 = 0.0011

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

q-0.4

-0.2

0

0.2

0.4

0.6

e L

Figure 5: Inflation level, output gap and exchange rate for different φ corresponding todistinct economies: Spain (φ = −0.0038, solid line), and Sweden (φ = 0.0011, dashedline). (Authors’ Matlab simulation)

As to the effects of forward guidance in a liquidity trap, as in the previous case, the

response of the three variables becomes sizable for values of q ≥ 0.6. However, under

liquidity trap, inflation in Sweden is characterized by a peak and then it does not move

substantially, while for Spain we observe that there is a trough (more or less when there

is the peak for Sweden), followed by a peak and then inflation remains positive. Output

gap moves in the same direction for both countries, while the real exchange rate depreci-

ates significantly but temporarily only for the case of a larger exchange rate pass-trough.

Overall, these results confirm that the exchange rate pass-through is a key variable.

4.3 Marginal Effects of the Exchange-Rate Pass Through and For-ward Guidance Duration

Here we study the effects of the exchange-rate pass-through as well as forward guidance

duration on the endogenous variables. Since analytical solutions become cumbersome,

7Moreover, if we calibrate the economy using the data for for Spain, we could show that forward guid-ance turns out to be more expansionary in closed economy.

13

for tractability, we first study closed-form solutions only for a version of the model with

contemporaneous Phillips curve (as in Bilbiie (2019)), then we examine a more general

version through numerical simulations.

4.3.1 Marginal Effects for the Contemporaneous Phillips Curve in State F

Following the methodology in Bilbiie (2019), and taking into account the expected values

for output gap, inflation and exchange rate, we first solve for the state F and then for the

state L. In state F we must solve the following system in xF , πF and eF , (for β = 0,

meaning that the Phillips curve is contemporaneous):

πF = κxF −φeF ,

xF = (1− p)qxF −σ−1(ρt− (1− p)qπF)+σ

−1ρt−δ ((1− p)qeF − eF)

ρt− (1− p)qπF = (1− p)qeF − eF

Solving the previous expression and remembering the relationship with πF , we get the

following pair of values for the state F :

πF =−(κδ −φ)

1−q [κ (δ +σ−1)−φ +1]ρt

xF =−[δ (1−q)−φσ−1q

](1−q)1−q [κ (δ +σ−1)−φ +1]

ρt

eF =−[1−q−qκσ−1]

(1−q)1−q [κ (δ +σ−1)−φ +1]ρt

We now expose the solutions obtained for state L using values given in state F :

πL = κ

q(1− p)

(1+κσ−1)

(1− p−σ−1 pκ)+

φ [p+(1− p)q][φ p+(1− p)]

xF

−q(1− p)

κ(σ−1φ +δ

)(1− p−σ−1 pκ)

+φ (1+φ)

[φ p+(1− p)]

eF

− φ

[φ p+(1− p)]ρL +

κ[δ (1− p)−σ−1φ p

](1− p−σ−1 pκ)

eL

xL =

(1− p−σ−1 pκ

)q(1− p)(1+κσ−1)

xL−[δ (1− p)−φσ−1σ−1 p

]q(1− p)(1+κσ−1)

eL +

(σ−1φ +δ

)(1+κσ−1)

eF

14

eL =[φ p+(1− p)]q(1− p)

(1+κσ−1)

[φ p+(1− p)]AρL

+q(1− p)q(1− p)

[(1+φ)

(1+κσ−1)− (σ−1φ +δ

)]− p

(σ−1φ +δ

)A

eF

−κ [p+(1− p)q]

(1− p−σ−1 pκ

)A

xL

where A=[φ p+(1− p)]q(1− p)

(1+κσ−1)−κ [p+(1− p)q]

[δ (1− p)−φσ−1 p

].

With a contemporaneous Phillips curve, the effect of forward guidance duration on

output gap is positive for a critical value of q:

∂πF

∂q=

q(κδ −φ)[κ(δ +σ−1)+1−φ

]1−q [κ (δ +σ−1)−φ +1]2 ρt > 0

∂xF

∂q=

φσ−1−[κ(δ +σ−1)+1−φ

][δ (1−q)2 +φσ−1q2]

(1−q)2 1−q [κ (δ +σ−1)−φ +1]2 ρt > 0,

∂eF

∂q=

κσ−11−q2 [κ (δ +σ−1)+1−φ]− (1−q)2 [κ (δ +σ−1)+1−φ

](1−q)2 1−q [κ (δ +σ−1)−φ +1]2 ρt ,

∂πF

∂φ=

2φ −κδ −qφ[κ(δ +σ−1)+1−φ

]1−q [κ (δ +σ−1)−φ +1]2 ρt > 0,

∂xF

∂φ= q

σ−11−q[κ(δ +σ−1)+1

]+δ (1−q)

(1−q)1−q [κ (δ +σ−1)−φ +1]2 ρt > 0,

∂eF

∂φ=

−q

1−q [κ (δ +σ−1)−φ +1]2 ρt < 0.

Under the condition (κδ −φ)> 0 which generally holds in our simulations with β =

0, any increase of q generates positive effects in state F on inflation and output gap; the

effect is ambiguous on the exchange rate. Taking into account the relationship of πF with

xF , we can conclude that forward guidance determines output and inflation expansion,

together with real depreciation.

Furthermore, any increase in φ generates positive response in state F of inflation, the

output gap but always a negative one of the exchange rate. This is straightforward from

observing equations (1)-(4).

4.3.2 Marginal Effects in a Liquidity Trap

We now turn to the initial model where the Phillips curve corresponds to equation (2).

Here we consider the model above described using the calibrations for two economies:

Sweden and Spain. We solve numerically the model and then compute the derivatives of

inflation, output gap and exchange rate with respect to q and φ respectively. This will

15

allow us to characterize analytically how exchange rate passthrough and duration of the

policy affect the transmission of forward guidance.

Marginal effect of forward guidance duration Here we study the effect of an in-

crease in the duration of forward guidance episod in state L depending on the openness

of the economy. We show numerically that, in Spain and Sweden, an increase in the du-

ration of the forward guidance regime, always yields higher inflation, a higher output gap

and the depreciation of the exchange rate, everything else being equal for any value of ω

(Figures 6a-6b) and q (Figure 7) in state L. For both economies, this effect is stronger the

longer the forward guidance, but an economy that is quite open (high ω) will experience

lower marginal effects. If we consider Spain for instance, when comparing the response

when the economy is closed (ω = 0) and q = 0.2 corresponding to Figure 6a, with the

case where the economy is closed and q = 0.8 corresponding to Figure 6b, the value is

almost 4 times bigger when the duration of forward guidance increases from 1 to 5 quar-

ters. This may suggest that there exists an adequate duration of the policy depending on

the size of the stimulus the policy maker is aiming at.

Marginal effect of pass-through of exchange rate to inflation Comparing both

countries, we can see that the variations of the exchange-rate pass-through produce dif-

ferent effects depending on the duration of forward guidance. With q = 0.2 (figure 6a)

output, inflation and exchange rate in Spain are decreasing in φ . With q = 0.8 (figure

6b), inflation and exchange rate are increasing, while output first decreases reaching a

minimum then it bounces back. On the other hand, we observe that in Sweden the three

variables are much less sensitive. Therefore, in state L, the marginal effect of the pass-

through depends strongly on the features of the economy, in particular on the elasticity of

the interest rate to the output gap in the IS curve, σ , which is higher in Spain compared

to Sweden (see Table 1).

From Figure 7, it turns out that in Spain whenever q > 0.3, a higher pass-through

reduces output, while a threshold value of 0.4 and 0.55 are those that determine a negative

response of inflation and the exchange rate respectively. A lower duration of forward

guidance do not affect the transmission of exchange rate to the economy. As to Sweden,

as already highlighted above, the effects are sensibly lower. The top panel of Figure 7

confirms that the expansionary effects of forward guidance increase in its duration.

16

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

410

-3

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

Sweden

Spain

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

(a) Derivatives in state L for q = 0.2

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Sweden

Spain

0 0.2 0.4 0.6 0.8 1-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

(b) Derivatives in state L for q = 0.8

Figure 6: Derivatives in state L for different openness of each economy for a given dura-tion of forward guidance regime.

17

0 0.2 0.4 0.6 0.8

q

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8

q

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.2 0.4 0.6 0.8

q

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8

q

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 0.2 0.4 0.6 0.8

q

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Sweden

Spain

0 0.2 0.4 0.6 0.8

q

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Figure 7: Derivatives in state L for different q with calibrations from Table 1 (authors’Matlab simulation).

5 Conclusion

This paper studies forward guidance in a theoretical DSGE for an advanced small open

economy. We show that the elasticity of inflation to the real exchange rate is a key variable

in determining the qualitative and quantitative response of the economy to a policy of fixed

interest rates. The expansionary effect of the policy is positively related to the exchange

rate pass-through and larger than in the closed economy counterpart because of a better

inflation-output trade-off and the exchange rate channel. These findings generally hold

also in the case in which forward guidance is implemented during a liquidity trap.

Our analysis suggests that when considering the possibility to adopt forward guid-

ance, central banks should take into account how the exchange rate channel impacts in

the policy transmission, as multiple equilibria and different responses could arise depend-

ing on the structure of the economy. These results might not necessarily be extended to

emerging market economies. Several extensions to our setup can be considered. First, we

do not analyze optimal forward guidance and in particular how it is related to the open

economy dimension. Second, with incomplete information set available to the central

bank, there might be an attenuation of the forward guidance puzzle. Third, forward guid-

ance could produce heterogeneous effects as shown by Ferrante and Paustian (2019) in a

HANK model. Finally, we have abstracted from fiscal shocks and on how the interaction

between monetary and fiscal policy modifies the transmission of forward guidance.

18

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